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D. GILL shell, the inequality is an empirically verifiable con- experiments which are not performed; and even if sequence of the idea that the outcome of one mea- we do that, why should we demand that these coun- surement on one system cannot depend on which terfactual objects satisfy the same kind of physical measurement is performed on the other. This idea, constraints as the factual objects? However, it is a called locality or, more precisely, relativistic local fact that prior to quantum physics, realism was a causality, is just one of the three principles. Its completely natural property of all physical theories. formulation refers to outcomes of measurements The concepts of realism and locality together are which are not actually performed, so we have to often considered as one principle called local realism. assume their existence, alongside of the outcomes Local realism is implied by the existence of local hid- of those actually performed: the principle of real- den variables, whether deterministic or stochastic. ism, or more precisely, counterfactual definiteness. In a precise mathematical sense, the reverse impli- Finally, we need to assume that we have complete cation is also true: local realism implies that we can freedom to choose which of several measurements to construct a local hidden variable (LHV) model for perform—this is the third principle, also called the the phenomenon under study. However, one likes to no-conspiracy principle or no super-determinism. think of this assumption (or pair of assumptions), (As we shall see, super-determinism is a conspira- the important thing to realize is that it is a com- torial form of determinism.) pletely unproblematic feature of all classical physi- We shall implement the freedom assumption as cal theories; freedom (no conspiracy) even more so. the assumption of statistical independence between The connection between Bell’s theorem and statis- the randomisation in a randomised experimental de- tical notions of causality has been noted many times sign, and the set of outcomes of each experimen- tal unit under all possible treatments. This set con- in the past. For instance, in a short note, Robins, sists of the “counterfactual” outcomes of those treat- VanderWeele and Gill (2015) derive Bell’s inequal- ments which were not actually applied, as well as ity using the statistical language of causal interac- the “factual” outcome belonging to the treatment tions. The causal graph (DAG) of observed and un- chosen by the randomisation. observed variables corresponding to a classical phys- By existence of the outcomes of not actually per- ical description of one run of a standard Bell exper- formed experiments, we mean their mathematical iment is given in Figure 1. Alice and Bob’s settings existence within some mathematical-physical theory are binary: they independently use randomisation (a of the phenomenon in question. So “realism” actu- coin toss) to choose between one of two settings on a ally refers to models of reality, not to reality itself. measurement device. The outcome of each measure- Moreover, it could be thought of as a somewhat ment is also binary. Observed variables are repre- idealistic position. If we already have an adequate sented by grey rectangles; unobserved (there is only mathematical physical model of reality, there would one, but of course it might be of arbitrarily complex not seem to be a pressing need to add into this the- nature) by a white oval. The validity of this causal ory some mathematical description of outcomes of model places restrictions on the joint distribution of

Fig. 1. A classical description of a Bell-CHSH type experiment entails the validity of the graphical model described by this simple causal graph. Rectangles: observed variables; ellipse: unobserved. Settings and outcomes are both binary. Experimental results arguably (via Bell’s theorem) show that the classical description has to be abandoned. The probability distribution of experimental data is far outside the class of probability distributions allowed by the model. STATISTICS, CAUSALITY AND BELL’S THEOREM 3 the observed variables; see, for instance, Ver Steeg get to see the value of A or A′, and either the value and Galstyan (2011). of B or B′. We can therefore determine the value In view of the experimental support for violation of just one of the four products AB, AB′, A′B, and ′ ′ 1 of Bell’s inequality, the present writer prefers to A B , each with equal probability 4 , for each row imagine a world in which “realism” is not a fun- of the table. Denote by AB obs the average of the damental principle of physics but only an emergent observed products of A hand iB (“undefined” if the ′ ′ property in the familiar realm of daily life. In this sample size is zero). Define AB obs, A B obs and way, we can keep quantum mechanics, locality and A′B′ similarly. h i h i h iobs freedom. This position does entail taking quantum ′ ′ randomness very seriously: it becomes an irreducible Fact 1. For any four numbers A, A , B, B each equal to 1, feature of the physical world, a “primitive notion”; ± ′ ′ ′ ′ it is not “merely” an emergent feature. He believes (1) AB + AB + A B A B = 2. that within this position, the measurement problem − ± (Schr¨odinger cat problem) has a decent mathemat- Proof. Notice that ical solution, in which causality is the guiding prin- ′ ′ ′ ′ ′ ′ ′ AB + AB + A B A B = A(B + B )+ A (B B ). ciple (Slava Belavkin’s “eventum mechanics”). − − Many practical minded physicists claim to be ad- B and B′ are either equal to one another or unequal. herents of the so-called Many Worlds interpretation In the former case, B B′ =0 and B + B′ = 2; in (MWI) of quantum mechanics. In the writer’s opin- the latter case B B′−= 2 and B + B′ = 0.± Thus, ion (but also of many writers on quantum founda- AB + AB′ + A′B −A′B′ equals± either A or A′, both tions), this interpretation also entails a rejection of of which equal −1, times 2. All possibilities lead “realism”, but now in a very strong sense: the reality to AB + AB′ +±A′B A′B±′ = 2.  of an actual random path taken by Nature through − ± space–time is denied. The only reality is the ensem- Fact 2. ble of all possible paths. Devilish experiments lead ′ ′ ′ ′ (2) AB + AB + A B A B 2. to dead cats turning up on some paths, and alive h i h i h i − h i≤ cats on others. According to MWI, the only real- Proof. By (1), ity is the quantum wave-function. The reality of the ′ ′ ′ ′ AB + AB + A B A B death (or not) of the cat is an illusion. h i h i h i − h i ′ ′ ′ ′ = AB + AB + A B A B [ 2, 2].  2. BELL’S INEQUALITY h − i∈ − Formula (2) is known as the CHSH inequality To begin with, I will establish a new version of the (Clauser et al. (1969)). It is a generalisation of the famous Bell inequality (more precisely: Bell-CHSH original Bell (1964) inequality. inequality). My version is not an inequality about When N is large one would expect AB obs to theoretical expectation values, but is a probabilis- be close to AB , and the same for theh otheri three tic inequality about experimentally observed aver- averages ofh observedi products. Hence, equation (2) ages. Probability derives purely from randomisation should remain approximately true when we replace in the experimental design. the averages of the four products over all N rows Consider a spreadsheet containing an N 4 ta- with the averages of the four products in each of ble of numbers 1. The rows will be labelled× by an four disjoint sub-samples of expected size N/4 each. index j = 1,...,N± . The columns are labelled with ′ ′ The following theorem expresses this intuition in a names A, A , B and B . I will denote the four num- ′ precise and useful way. Its straightforward proof, bers in the jth row of the table by A , A , B and j j j given in the Appendix, uses two Hoeffding (1963) B′ . Denote by AB = (1/N) N A B , the aver- j h i j=1 j j inequalities (exponential bounds on the tail of bi- age over the N rows of the productP of the elements in nomial and hypergeometric distributions) to prob- the A and B columns. Define AB′ , A′B , A′B′ h i h i h i abilistically bound the difference between AB obs similarly. and AB , etc. h i Suppose that for each row of the spreadsheet, two h i fair coins are tossed independently of one another, Theorem 1. Given an N 4 spreadsheet of ×′ ′ independently over all the rows. Suppose that de- numbers 1 with columns A, A , B and B , sup- pending on the outcomes of the two coins, we either pose that,± completely at random, just one of A and 4 R. D. GILL

A′ is observed and just one of B and B′ are observed blame on realism, and not in the weak sense of the in every row. Then, for any η 0, Copenhagen interpretation which is a kind of dog- ≥ ′ ′ matic assertion that it doesn’t make any sense to ask Pr( AB obs + AB obs + A B obs h i h ′ i ′ h i “what is actually going on behind the scenes”, but (3) A B 2+ η) − h iobs ≤ in a more positive sense: the positive assertion that − 2 quantum randomness is both real and fundamental. 1 8e N(η/16) . ≥ − In classical physics, randomness is merely the result Traditional presentations of Bell’s theorem derive of dependence on uncontrollable initial conditions. the large N limit of this result. If for N , experi- →∞ Variation in those conditions, or uncertainty about mental averages converge to theoretical mean values, them, leads to variation, or uncertainty, in the fi- then by (3) these must satisfy nal result. However, there is no such explanation ′ ′ ′ ′ for quantum randomness. Quantum randomness is (4) AB lim + AB lim + A B lim A B lim 2. h i h i h i − h i ≤ intrinsic, nonclassical, irreducible. It is not an emer- Like (2), this inequality is also called the CHSH in- gent phenomenon. It is the bottom line. It is a fun- equality. damental feature of the fabric of reality. I conclude this section with an open problem. An For present purposes, we do not need to under- analysis by Vongehr (2013) of the original Bell in- equality, which is “just” the CHSH inequality in the stand any of the quantum mechanics behind (6): we situation that one of the four correlations is identi- just need to know the specific statistical predictions cally equal to 1, suggests that the following con- which follow from a particular model in quantum jecture might be± true. I come back to this in the last physics called the EPR-B model. The initials refer section of the paper. here to the celebrated paradox of Einstein, Podolsky and Rosen (1935) in a version introduced by Bohm Conjecture 1. Under the assumptions of The- (1951). The EPR-B model predicts the statistics of orem 1, measurements of spin on each of an entangled pair ′ ′ ′ ′ Pr( AB obs + AB obs + A B obs A B obs > 2) of spin-half quantum systems in the singlet state. (5) h i h i h i − h i Fortunately, we do not need to understand any of 1 . ≤ 2 these words in order to understand what an EPR-B experiment looks like (see Figure 1 again). 3. BELL’S THEOREM In one run of this stylised experiment, two par- Both the original Bell inequality, and Bell-CHSH ticles are generated together at a source, and then inequality (4), can be used to prove Bell’s theorem: travel to two distant locations. Here, they are mea- quantum mechanics is incompatible with the princi- sured by two experimenters Alice and Bob. Alice ples of realism, locality and freedom. If we want to and Bob are each in possession of a measurement hold on to all three principles, quantum mechanics apparatus which can “measure the spin of a parti- must be rejected. Alternatively, if we want to hold cle in any chosen direction”. Alice (and similarly, on to quantum theory, we have to relinquish at least Bob) can freely choose (and set) a setting on her one of those three principles. measurement apparatus. Alice’s setting is an arbi- An executive summary of the proof of Bell’s the- trary direction in real three-dimensional space rep- orem consists purely of the following one-liner: cer- resented by a unit vector a. Her apparatus will then tain models in quantum physics, referring to an ex- register an observed outcome 1 which is called the periment with the layout of Figure 1, predict observed spin of Alice’s particle± in direction a. At ′ ′ ′ ′ b AB lim + AB lim + A B lim A B lim the same time, far away, Bob chooses a direction (6) h i h i h i − h i and also gets to observe an outcome 1. This is re- = 2√2. peated many times—the complete experiment± will More details will be given in a moment. consist of a total of N runs. We will imagine Alice If we accept quantum mechanics, should we re- and Bob repeatedly choosing new settings for each ject locality, realism, or freedom? Almost no-one is new run, in the same fashion as in Section 2: each prepared to abandon freedom. It seems to be a mat- tossing a fair coin to make a binary choice between ′ ter of changing fashion whether one blames local- just two possible settings, a and a for Alice, b and ity or realism. I will argue that we must place the b′ for Bob. STATISTICS, CAUSALITY AND BELL’S THEOREM 5

First, we will complete our description of the measured in Bob’s wing. Thus, for each run there is quantum mechanical predictions for each run sepa- a suite of potential outcomes A, A′, B and B′, but rately. For pairs of particles in the singlet state, the only one of A and A′, and only one of B and B′ ac- prediction of quantum mechanics is that in what- tually gets to be observed. By freedom, the choices ever directions Alice and Bob perform their mea- are statistically independent of the actual values of surements, their outcome 1 is completely random, the four. that is, both marginal distributions± are uniform. I will assume furthermore that the suite of coun- The outcomes are not, however, independent. They terfactual outcomes in the jth run does not actually are correlated, with correlation depending on the depend on which particular variables were observed two settings. To be precise, the expected value of the in previous runs. This memoryless assumption can product of the outcomes is equal to a b = cos(θ) be completely avoided by using the martingale ver- where θ is the angle between the two− directions.· − sion of Hoeffding’s inequality, Gill (2003). But the With this information, we can write down the 2 present analysis is already applicable if we imagine 2 table for the joint probability distribution of the× N copies of the experiment each with only a sin- outcomes at the two locations, given two settings gle run, all being done simultaneously in different differing in direction by the angle θ: laboratories. The assumptions of realism, locality and freedom +1 −1 have put us firmly in the situation of the previous +1 1 (1 cos(θ)) 1 (1 + cos(θ)) section. Therefore, by Theorem 1, the four sample 4 − 4 1 1 (1 + cos(θ)) 1 (1 cos(θ)) correlations (empirical raw product moments) sat- − 4 4 − isfy (3). Both marginals of the table are uniform. The ex- Let us contrast this prediction with the quan- pectation of the product of the outcomes equals the tum mechanical predictions obtained with a certain clever selection of directions. We will take the four probability that they are equal minus the probabil- ′ ′ 2 2 vectors a, a , b and b to lie in the same plane. ity they are different 4 (1 cos(θ)) 4 (1 + cos(θ)) = ′ cos(θ). Physicists use− the word− “correlation” to It is then enough to specify the angles α, α , β, − β′ [0, 2π] which they make with respect to some refer to the raw (uncentered, unnormalised) prod- ∈ uct moment but in this case the physicist’s and the fixed vector in this plane. Consider the choice α = 0, α′ = π/2, β = 5π/4, β′ = 3π/4; see Figure 2. The statistician’s correlation coincide. ′ ′ As mentioned before, Alice and Bob now perform differences α β , α β , α β are all equal to π π/4: these| − pairs| | of− angles| | − are| only 45 degrees N runs of the experiment according to the following ± randomised experimental design. Alice has fixed in away from being opposite to one another; the corre- advance two particular directions a and a′; Bob has sponding measurements are quite strongly positively ′ correlated. On the other hand, α′ β′ = π/4: these fixed in advance two particular directions b and b . | − | In each run, Alice and Bob are each sent one of a two angles are 45 degrees away from being equal new pair of particles in the singlet state. While their and the corresponding measurements are as strongly particles are en route to them, they each toss a fair ′ coin in order to choose one of their two measurement ′ a directions. In total N times, Alice observes either b A = 1 or A′ = 1 say, and Bob observes either B = ±1 or B′ = ±1. At the end of the experiment, a four± “correlations”± are calculated: the four sample ′ ′ ′ ′ averages of the products AB, AB , A B and A B . b Each correlation is based on a different subset of runs, of expected size N/4, determined by the 2N Fig. 2. Four measurement directions, all in the same plane. ′ fair coin tosses. Alice’s settings are the two orthogonal directions a, a , and b b′ Under realism we can imagine, for each run, along- Bob’s settings are the orthogonal , . Relative to one an- other, the pairs are arranged so that a′ and b′ are close to side of the outcomes of the actually measured pair pointing in the same direction, while at the same time the of variables, also the outcomes of the not measured other three pairs of one of Alice’s and one of Bob’s settings pair. Under locality, the outcomes in Alice’s wing (a and b, a and b′, a′ and b) are all equally close to pointing cannot depend on the choice of which variable is in opposite directions. 6 R. D. GILL anti-correlated, as the other pairs were strongly An important difference between the idealisation correlated. Three of the correlations are equal to and the truth concerns the picture of Alice and Bob cos(3π/4) = ( 1/√2) = 1/√2 and the fourth is repeating some actions N times with N fixed in equal− to cos(−π/−4) = 1/√2. Thus, we would ex- advance. The experimenters do not control when a − − pect to see, up to statistical variation, pair of photons will leave the source nor how many ′ ′ ′ ′ AB + AB + A B A B times this happens. Even talking about “pairs of h iobs h iobs h iobs − h iobs photons” is using classical physical language which 4/√2 = 2√2 2.828, can be acutely misleading. In actual fact, all we ob- ≈ ≈ serve are individual detection events (time, current cf. (6). By Tsirelson’s inequality (Cirel’son (1980)), setting, outcome) at each of the two detectors, that this is actually the largest absolute deviation from is, at each measurement apparatus. the CHSH inequality which is allowed by quantum mechanics. Complicating this still further is the fact that Many experiments have been performed confirm- many particles fail to be detected at all. One could ing these predictions. Two particularly notable ones say that the outcome of measuring one particle is not binary but ternary: +, , or no detection. If are those of Aspect, Dalibard and Roger (1982) in − Orsay, Paris, and of Weihs et al. (1998) in Innsbruck neither particle of a pair is detected, then we do not (later I will discuss two of the most recent). even know there is a pair at all. N was not only In these experiments, the choices of which direc- not fixed in advance: it is not even known. The data tion to measure were not literally made with coin cannot be summarised in a list of pairs of settings tosses, but by rather more practical physical sys- and pairs of outcomes (whether binary of ternary), tems. In Alain Aspect’s Orsay and Gregor Weihs’ but consists of two lists of the random times of def- Innsbruck experiments, the separation between the inite measurement outcomes in each wing of the ex- locations of Alice and Bob was large; the time it periment together with the settings in force at the took from initiating the choice of random direction time of the measurements. The settings are being to measure to completion of the measurement was extremely rapidly, randomly switched, between the small: so small, that Alice’s measurement is com- two alternative values. When detection events occur plete before a signal traveling at the speed of light close together in time they are treated as belonging could possibly transmit Bob’s choice to Alice’s loca- to a pair of photons. tion. However, note that this depends on what one In Weihs’ experiment, only 1 in 20 of the events considers to be the time each randomisation starts in each wing of the experiment seemed to be paired happening. Weihs’ experiment improves on Aspect’s with an event in the other. If all detections corre- in this respect. spond to emissions of pairs from the source, then The data gathered from the Innsbruck experi- for every 400 pairs of photons, just one pair leads to ment is available online. It had N 15,000; and a paired event, 2 19 lead to unpaired events, and ′ ′ ≈ ′ ′ × found AB obs + AB obs + A B obs A B obs = the remaining 361 to no observed event at all. 2.73 h0.022,i theh statisticali accuracyh i − (standard h i de- We will return to the issue of whether the idealised viation)± following from a standard delta-method picture of N pairs of particles, each separately be- calculation assuming i.i.d. observations per setting ing measured, each particle in just one of two ways, pair. The reader can check that this corresponds is really appropriate, in a later section; we will also to accuracy obtained by a standard computation take a look then at two more, very recent, exper- using binomial variances of the counts for each of iments. However, the point is that quantum me- the four roughly equal sub-samples. By (3), un- chanics does seem to promise that experiments of der realism, locality and freedom, the chance that this nature could in principle be done, and if so, ′ ′ ′ ′ AB obs + AB obs + A B obs A B obs would ex- there seems no reason to doubt they could violate ceedh i 2.73 ish lessi thanh 10−12i. −h i the CHSH inequality. Three correlations more or less The experiment deviates in several ways from equal to 1/√2 and one equal to 1/√2 have been − what has been described so far, and I will summarise measured in the lab. Not to mention that the whole them here. curve cos(θ) has been experimentally recovered. An unimportant difference is the physical system Right− now the situation is that at least four ma- used: polarisation of entangled photons rather than jor experimental groups (Singapore, Brisbane, Vi- spin of entangled spin-half particles (e.g., electrons). enna, Illinois) seem to be vying to be the first to STATISTICS, CAUSALITY AND BELL’S THEOREM 7 perform a successful and completely “loophole-free” Whether it needed external support or not, the no- experiment, predictions being that this is no more tion of counterfactual definiteness is nothing strange than five years away (cf. Marek Zukowski,˙ quoted in in all of physics (prior to the invention of quantum Merali (2011)). It will be a major achievement, the mechanics). It comes for free with a deterministic crown of more than fifty years’ labour. view of the world as a collection of objects blindly obeying definite rules. 4. REALISM, LOCALITY, FREEDOM Instead of assuming quantum mechanics and de- This section and the next are about metaphysics riving counterfactual definiteness, Bell turned the and can safely be skipped by the reader impatient EPR argument on its head. He assumes three prin- to learn more about statistical aspects of Bell ex- ciples which Einstein would have endorsed anyway, periments. and uses them to get a contradiction with quantum The EPR-B correlations have a second message mechanics; and the first is counterfactual definite- beyond the fact that they violate the CHSH inequal- ness. We must first agree that though, say, only A and B are actually measured in one particular run, ity. They also exhibit perfect anti-correlation in the ′ ′ case that the two directions of measurement are ex- still, in a mathematical sense, A and B also exist actly equal—and perfect correlation in the case that (or at least may be constructed) alongside of the they are exactly opposite. This brings us straight to other two; and moreover, they may be thought to the EPR argument not for the nonlocality of quan- be located in space and time just where one would tum mechanics, but for the incompleteness of quan- imagine. Only after that does it make sense to dis- tum mechanics. cuss locality: the assumption that which variable is Einstein, Podolsky and Rosen (1935) were re- being observed at Alice’s location does not influence vulsed by the idea that the “last word” in physics the values taken by the other two at Bob’s location. would be a “merely” statistical theory. Physics Having assumed realism and locality, we can bring should explain why, in each individual instance, the freedom assumption into play. As we have seen, what actually happens does happen. The belief that it allowed us to analyse our experiment with clas- every “effect” must have a “cause” has driven West- sical probability tools based on a classically ran- ern science since Aristotle. Now according to the sin- domised design. Some writers like to associate the glet correlations, if Alice were to measure the spin freedom assumption with the free will of the exper- of her particle in direction a, it is certain that if imenter, others with the existence of “true” ran- Bob were to do the same, he would find exactly domness in other physical processes: either way, the opposite outcome. Since it is inconceivable that one metaphysical assumption is justified by another. Alice’s choice has any immediate influence on the I would rather see it in a practical way: we under- particle over at Bob’s place, it must be that the stand pseudo-randomness very well, and its princi- outcome of measuring Bob’s particle in the direc- ples underly coin tossing just as much as pseudo ran- tion a is predetermined “in the particle” as it were. dom generators. We use randomisation effectively in The measurement outcomes from measuring spin in all kinds of contexts (randomised algorithms, ran- all conceivable directions on both particles must be domised clinical trials, randomised designs). Do we predetermined properties of those particles. The ob- really want to believe that the observed correlations served correlation is merely caused by their origin at 0.7071 (three positive, one negative) come about ± a common source. through a physical mechanism by which the out- Thus Einstein used locality, together with the pre- comes of two coin tosses and two polarisation mea- dictions of quantum mechanics itself, to infer real- surements are all exquisitely dependent on one an- ism or counterfactual definiteness in the strong sense other through all four being jointly pre-determined that the outcomes of measurements on physical sys- by events in the deep past? When such a hypothe- tems are predefined properties of those systems, car- sis is otherwise completely unnecessary? (Otherwise, ried in them, and merely revealed by the act of we never see spatial-temporal correlations follow- measurement. From this, he argued the incomplete- ing this sign pattern, larger in absolute value than ness of quantum mechanics—it describes some ag- 0.5). A mechanism which is completely unknown? gregate properties of collectives of physical systems, A mechanism which ensures in effect that Alice’s but does not even deign to talk about physically photon knows how Bob’s photon is being measured? definitely existing properties of individual systems. Yet, a mechanism which cannot make any of those 8 R. D. GILL correlations larger in absolute value than 0.7071, comfort is provided) without having a prior frame- though if it really were the case that Alice’s photon work in which to interpret the data of experience knows Bob’s setting, three positive and one negative and experiment. It seems that we have modules for correlation 1.0 could have been achieved. algebra and modules for geometry: basic notions I think that± Occam’s razor tells us to discard this of number and of space. Most interestingly in the flavour of super-determinism, also known as conspir- present context, we also have modules for causality. acy. In fact, to abandon freedom means to abandon We distinguish between objects and agents (we learn science: we may discard all empirical (observational) that we ourselves are agents). Objects are acted data. Everything is explained but nothing can be on by agents. Objects have continuous existence in predicted. space–time, they are local. Agents can act on ob- Keeping freedom, we have to make a choice be- jects, also at a distance. Together this seems to me tween two other inconceivable possibilities: do we to be a built-in assumption of determinism: we have reject locality, or do we reject realism? been created (by evolution) to operate in an Aris- Here, I would like to call on Occam’s principle totelian world, a world in which every effect has a again. Suppose realism is true. Instead of invok- cause. ing the fact that a collection of four coin toss out- The argument (from physics, and by Occam’s comes and photo-detector clicks were jointly prede- razor, not from neuroscience) for abandoning re- termined in the deep past, we now have to invoke alism is made eloquently by Boris Tsirelson in instantaneous communication across large distances an internet encyclopaedia article on entanglement of the outcomes of these processes, by as yet un- (Citizendium: entanglement). It was Tsirelson from known processes, and again with only the extremely whom I borrowed the terms counterfactual definite- subtle and special effects which quantum mechanics ness, relativistic local causality, and no-conspiracy. seems to predict. Alice cannot communicate with He points out that it is a mathematical fact that Bob through this phenomenon. There is no observ- quantum physics is consistent with relativistic local able action-at-a-distance. The surface predictions of causality and with no-conspiracy. In all of physics, quantum mechanics are perfectly compatible with there is no evidence against either of these two prin- relativistic causality. It is only when we hypothesise ciples. a hidden layer that we run into difficulties. I would like to close this section by mention- It seems to me that we are pretty much forced ing a beautiful paper by Masanes, Acin and Gisin into rejecting realism, which, please remember, is (2006) who argue in a very general setting (i.e., actually an idealistic concept: outcomes “exist” of not assuming quantum theory, or local realism, or measurements which were not performed. However, anything) that quantum nonlocality, by which they I admit it goes against all instinct. In the case of mean the violation of Bell inequalities, together equal settings, how can it be that the outcomes are with nonsignalling, which is the property that the marginal probability distribution seen by Alice of A equal and opposite, if they were not predetermined ′ at the source? does not depend on whether Bob measures B of B , Though it is perhaps only a comfort blanket, I together imply indeterminism: that is to say: that would like here to appeal to the limitations of our the world is stochastic, not deterministic. own brains, the limitations we experience in our “un- 5. RESOLUTION OF THE MEASUREMENT derstanding” of physics due to our own rather spe- PROBLEM cial position in the universe. In philosophy, this no- tion is called embodied cognition. There is also hard The measurement problem, also known as Schr¨odin- empirical evidence for this idea. ger’s cat problem, is the problem of how to recon- According to cognitive scientists (see, for instance, cile two apparently mutually contradictory parts of Spelke and Kinzler (2007)), our brains are at birth quantum mechanics. When a quantum system is iso- hardwired with various basic conceptions about the lated from the rest of the world, its quantum state world. These “modules” are called systems of core (a vector, normalised to have unit length, in Hilbert knowledge. The idea is that we cannot acquire new space) evolves unitarily, deterministically. When we knowledge from our sensory experiences (including look at a quantum system from outside, by mak- learning from experiments: we cry, and food and/or ing a measurement on it in a laboratory, the state STATISTICS, CAUSALITY AND BELL’S THEOREM 9 collapses to one of the eigenvectors of an operator has long worked with arbitrary algebras of observ- corresponding to the particular measurement, and ables, not necessarily the full algebra of a specific it does so with probabilities equal to the squared Hilbert space. With respect to those traditional ap- lengths of the projections of the original state vector proaches the only novelty is to suppose that the uni- into the eigenspaces. Yet the system being measured tary evolution when restricted to the sub-algebra is together with the measurement apparatus used to not invertible. It is an endomorphism, not an iso- probe it form together a much larger quantum sys- morphism. There is an arrow of time. tem, supposedly evolving unitarily and determinis- It turns out that the theory is mathematically tically in time. equivalent to important versions of the continuous Accepting that quantum theory is intrinsically spontaneous localisation (CSL) model, a way to stochastic, and accepting the reality of measure- solve the measurement problem by adding an ex- ment outcomes, led Belavkin (2000) to a mathe- plicit stochastic collapse term to the Schr¨odinger matical framework which he called eventum me- equation (Initially, the two theories seem quite dif- chanics which (in my opinion) indeed reconciles the ferent in nature). The problem of crafting a rela- two faces of quantum physics (Schr¨odinger evolu- tivistically invariant version of CSL remained open tion, von Neumann collapse) by a most simple de- for many years (and was a major obstruction to vice. Moreover, it is based on ideas of causality with its acceptance) yet just recently this problem has respect to time. I have attempted to explain this been solved by Bedingham (2011). See Pearle (1997, model in as simple terms as possible in Gill (2009). 2012) for further details. The following words will only make sense to those CSL has been eloquently championed over the with some familiarity with quantum mechanics. years by Philip Pearle and I refer the reader to his The idea is to model the world in the conventional many works, in particular the two just cited, both way with a Hilbert space, a quantum state on that explaining CSL and explaining why it does solve the space, and a unitary evolution. Inside this frame- measurement problem, while MWI does not. work, we look for a collection of bounded operators on the Hilbert space which all commute with one an- 6. LOOPHOLES other, and which are causally compatible with the unitary evolution of the space, in the sense that they In real world experiments, the ideal experimen- all commute with past copies of themselves (in the tal protocol of particles leaving a source at definite Heisenberg picture, one thinks of the quantum ob- times, and being measured at distant locations ac- servables as changing, the state as fixed; each ob- cording to locally randomly chosen settings cannot servable corresponds to a time indexed family of be implemented. bounded operators). We call this special family of Experiments have been done with pairs of entan- operators the beables: they correspond to physical gled ions, separated only by a short distance. The properties in a classical-like world which can co- measurement of each ion takes a relatively long time, exist, all having definite values at the same time, and but at least it is almost always successful. Such ex- definite values in the past too. The state and the uni- periments are obviously blemished by the so-called tary evolution together determine a joint probabil- communication or locality loophole. Each particle ity distribution of these time-indexed variables, that can know very well how the other one is being mea- is, a stochastic process. At any fixed time, we can sured. condition the state of the system on the past trajec- Many very impressive experiments have been per- tories of the beables. This leads to a quantum state formed with pairs of entangled photons. Here, the over all bounded operators which commute with all measurement of each photon can be performed very the beables. rapidly and at huge distance from one another. How- The result is a theory in which the deterministic ever, many photons fail to be detected at all. For and stochastic parts of traditional quantum theory many events in one wing of the experiment, there is are combined into one harmonious whole. In fact, often no event at all in the other wing, even though the notion of restricting attention to a sub-class of the physicists are pretty sure that almost all detec- all observables goes back a long way in quantum tion events do correspond to (members of) entangled theory under the name super-selection rule; and ab- pairs of photons. This is called the detection loop- stract quantum theory (and quantum field theory) hole. Popularly it is thought to be merely connected 10 R. D. GILL to the efficiency of photo-detectors and that it will of the probability that Alice sees an outcome given be easily overcome by the development of better and Bob sees an outcome (and vice versa). It is not to better photo-detectors. Certainly that is necessary, be confused with “detector efficiency”. It turns out ′ but not sufficient, as I will explain. that the (sharp) bound on AB lim + AB lim + ′ ′ ′ h i h i In Weihs’ experiment mentioned earlier, only 1 in A B lim A B lim set by local realism is no longer 20 of the events in each wing of the experiment is h2 as ini (4−), h but 2+i δ, where δ = δ(γ) = 4(γ−1 1). paired with an event in the other wing. Thus, of As long as γ 1/√2 0.7071, the bound− 2 + δ every 400 pairs of photons—if we assume that de- is smaller than 2≥√2. Weihs’≈ experiment has an effi- tection and nondetection occur independently of one ciency of 5%. If only we could increase it to above another in the two wings of the experiment—only 1 71% and simultaneously keep the state and mea- pair results in a successful measurement of both the surements close to perfection, we could have defini- photons; there are 19 further unpaired events in each tive experimental proof of Bell’s theorem. wing of the experiment; and there were 361 pairs of This would be correct for a “clocked” experi- photons not observed at all. ment. Suppose now particles determine themselves Imagine (anthropocentrically) classical particles the times that they are measured. Thus, a local about to leave the source and aiming to fake the realist pair of particles trying to fake the singlet singlet correlations. If they are allowed to go unde- correlations could arrange between themselves that tected often enough, they can engineer any correla- their measurement times are delayed by smaller or tions they like, as follows. Consider two new photons greater amounts depending on whether the setting about to leave the source. They agree between one they see at the detector is the setting they want another with what pair of settings they would like to see, or not. It turns out that this gives our de- to be measured. Having decided on the desired set- vious particles even more scope for faking correla- ting pair, they next generate outcomes 1 by draw- tions. Larsson and Gill (2004) called this the coin- ing them from the joint probability distribution± of cidence loophole, and derived the sharp bound on outcomes given settings, which they want the ex- ′ ′ ′ ′ AB lim + AB lim + A B lim A B lim set by lo- perimenter to see. Only then do they each travel to − calh realismi h is 2+iδ, whereh nowi δ−= h δ(γ)i = 6(γ 1 1). their corresponding detector. There, each particle As long as γ 3(1 1/√2) 0.8787, the bound− 2+δ compares the setting it had chosen in advance with is smaller than≥ 2√−2. We need≈ to get experimental the setting chosen by Alice or Bob. If they are not efficiency above 88%, and keep everything else close the same, it decides to go undetected. With proba- bility 1/4 we will have successful detections in both to perfect at the very limits allowed by quantum wings of the experiment. For those detections, the physics. pair of settings according to which the particles are How far is there still to go? In 2013, the Vienna being measured is identical to the pair of settings group published a paper in the journal Nature en- they had aimed at in advance. titled “Bell violation using entangled photons with- This example illustrates that if one wants to ex- out the fair-sampling assumption” (Giustina et al. perimentally prove a violation of local realism with- (2013)). The authors write “this is the very first time out making the untestable statistical assumption of that an experiment has been done using photons “missing at random”, known as the fair-sampling which does not suffer from the detection loophole”, assumption in this context, one has to put limits and moreover, the experiment “makes the photon on the amount of “nondetection”. There is a long the first physical system for which each of the main history and big literature on this topic. I will just loopholes has been closed, albeit in different exper- mention one of such results. iments”. Larsson (1998, 1999) has proved variants of the It was however rapidly pointed out that the ex- CHSH inequality which take account of the possi- periment was actually vulnerable to the coincidence bility of nondetections. The idea is that under lo- loophole, not “just” to the detection loophole. Now, cal realism, as the proportion of “missing” measure- it actually should be possible to simply re-analyse ments increases from zero, the upper bound “2” in the data from that experiment, defining coincidences the CHSH inequality (4) increases, too. We intro- with respect to an externally defined lattice of time duce a quantity γ called the efficiency of the ex- intervals instead of relative to observed detection periment: this is the minimum over all setting pairs times only. Ideally, this will only slightly increase STATISTICS, CAUSALITY AND BELL’S THEOREM 11 the “singles rate” and slightly decrease the num- in an experiment by Zeilinger. I am not however con- ber of coincidences, thereby slightly decreasing both vinced by this idea: though Alice’s setting choice is size and statistical significance of the Bell violation, triggered by a photon which cannot yet have inter- but hopefully without altering the substantive con- acted with Bob’s measurement device or the source, clusion. A more stringent re-analysis of the data still the setting itself is also partially determined (Larsson et al. (2013)) has confirmed that the initial by Alice’s detection apparatus, and it certainly has. claims were justified. And if there is dependence between subsequent set- In the meantime, exploiting this gap between re- tings on Alice’s side, then on Bob’s side it soon sults and claims, a consortium led by researchers becomes possible to predict future settings (this is from Illinois have published their own experimental known as the memory loophole). results, also reporting that theirs is “the first exper- In my opinion, we have to rule out conspiracy (en- iment that fully closes the detection loophole with sure freedom) by choosing settings by a cascade of photons, which are then the only system in which classical randomness (coins, pseudo RNGs, etc.). We both loopholes have been closed, albeit not simulta- can never logically rule out conspiratorial super de- neously” (Christensen et al. (2013)). They used the terminism, but we can make appeal to this escape Larsson and Gill (2004) inequality. clause ludicrous. Is this just a question of prestige? No: various new quantum technologies depend on quantum entangle- 7. BELL’S THEOREM WITHOUT ment, and in particular, various cryptographic com- munication protocols are not secure as long as it is INEQUALITIES possible to “fake” violation of Bell inequalities with In recent years new proofs of Bell’s theorem have classical systems. been invented which appear to avoid probability It is logically possible that quantum mechanics or statistics altogether, such as the famous GHZ itself could prevent one ever from performing a (Greenberger, Horne, Zeilinger) proof. Experiments both successful and loophole-free experiment. Quan- have already been done implementing the set-up tum uncertainty relations could in principle pre- of these proofs, and physicists have claimed that vent the creation of a multipartite quantum sys- these experiments prove quantum—nonlocality by tem, whose components can be measured in well- the outcomes of a finite number of runs: no statis- separated space–time regions, while simultaneously tics, no inequalities (yet their papers do exhibit error those components are in the required joint entangled bars). state. I christened this possibility “Bell’s fifth posi- Such a proof runs along the following lines. Sup- tion” in Gill (2003). Here, I just mention that the pose local realism is true. Suppose also that some possibility had already been championed for many event is certain. Suppose that it then follows from years by Emilios Santos, whose paper Santos (2005) local realismA that another event has probability is well worth reading. B On the other hand, continuous improvement of zero, while under quantum mechanics it can be ar- ranged that the same event has probability one. experimental techniques over more than fifty years B has seen continuous pushing of detection efficiency Paradoxical, but not a contradiction in terms: the catch is that events and are events under dif- toward the critical boundaries, without any attenua- A B tion of the quantum correlations of the singlet state. ferent experimental conditions: it is only under local realism and freedom that the events and can Now that we are getting very close indeed to the A B boundary, it would seem very unlikely that we won’t be situated in the same sample space. Freedom is be able to go past it. needed here to equate the probabilities of observ- I conclude this section with mention of some re- able events with those of unobservable events, just cent work on the conspiracy loophole. Recently, Gal- as in our own proof of Bell’s theorem. We need to licchio, Friedman and Kaiser (2014) have made the be able to assume that the subset of runs of the ex- novel suggestion to rule out conspiracy by the exper- periment in which the events were observable are a imental device of choosing settings with the help of random sample of the set of all repetitions. detection times of photons arriving from widely sep- When we use randomisation in experimental de- arated, and very distant galaxies from the dawn of sign, we are assuming that randomisation is inde- time. This idea will probably be implemented soon pendent of pre-existing unobserved characteristics of 12 R. D. GILL the experimental units. Is it plausible that the out- can only give strong statistical evidence, at best, comes of coin tosses used to create a randomised ex- that the probability in question is very close to 1 perimental design were predetermined together with indeed. But actually experiments are never perfect properties of the experimental units under study, so and more likely is that after a number of repeti- that our random sub-samples of units being given tions, one discovers that A>B actually has posi- { } a particular treatment, are actually heavily biased tive probability—that event will happen a few times. with regard to the properties we measure on them? The experimenter cannot create the required quan- Of course, under a purely deterministic world view tum state exactly, measurements are not perfect. (super-determinism) everything that was ever go- Thus, the logical conclusion from the experiment is ing to happen was determined in advance, at the that nothing has been proved. dawn of creation. But even in a deterministic world, To be sure, one can give a proof of Bell’s theorem pseudo-randomness exists and is well understood. that the theory of quantum mechanics is in conflict As an example, consider the following scenario, with local realism, which relies only on logic, not generalizing the Bell-CHSH scenario to the situation on probability. But if we want to use the set-up of where the outcome of the measurements on the two the proof as a set-up for an experiment, we move particles is not binary, but an arbitrary real number. to a different ball-park. We want to perform an ex- This situation has been studied by Zohren and Gill periment which gives strong evidence that nature is (2008), Zohren et al. (2010). incompatible with local realism. It turns out that Just as before, settings are chosen at random in whatever experimental set-up we take, we will nec- the two wings of the experiment. Under local realism essarily find ourselves explicitly or implicitly in the we can introduce variables A, A′, B and B′ repre- business of statistically proving violation of inequal- senting the outcomes (real numbers) in one run of ities, as the next section will make clear. the experiment, both of the actually observed vari- ables, and of those not observed. 8. BETTER BELL INEQUALITIES It turns out that it is possible under quantum ′ Why all the attention to the CHSH inequality? mechanics to arrange that Pr B A = Pr A There are others around, aren’t there? And are there ′ { ≤′ }′ { ≤ B = Pr B A =1 while Pr B A =0. On alternatives to “inequalities” altogether? I will ar- } { ≤ } { ≤ } ′ the other hand, under local realism, Pr B A = gue here that the whole story is “just” a collection ′ { ≤ }′ Pr A B = Pr B A = 1 implies Pr B of inequalities, and the reason behind this can be ′ { ≤ } { ≤ } { ≤ A = 1. expressed in a simple geometric picture. } Note that the four probability measures un- In a precise sense, the CHSH inequality is the only der which, under quantum mechanics, Pr A B , Bell inequality worth mentioning in the scenario of ′ ′ ′ ′ { ≤ } Pr A B , Pr A B , Pr A B are defined, two parties, two measurements per party, two out- { ≥ } { ≥ } { ≥ } refer to four different experimental set-ups, accord- comes per measurement. Let us generalise this sce- ing to which of the four pairs (A, B), etc. we are nario and consider p parties, each choosing between measuring. one of q measurements, where each measurement The experiment to verify these quantum mechan- has r possible outcomes (further generalisations are ical predictions has not yet been performed though possible to unbalanced experiments, multi-stage ex- some colleagues are interested. Interestingly, though periments, and so on). I want to explain why CHSH it requires a quantum entangled state, that state plays a very central role in the 2 2 2 case, and should not be the maximally entangled state (the why in general, generalised Bell inequalities× × are all amount of entanglement of a state can be quantified there is when studying the p q r case. The short in many ways, for instance through entropy notions, answer is: these inequalities are× the× bounding hyper- but it would take us too far into the quantum for- planes of a convex polytope of “everything allowed malism to explain that here). Maximal “quantum by local realism”. The vertices of the polytope are nonlocality” is quite different from maximal entan- deterministic local realistic models. An arbitrary lo- glement. And this is not an isolated example of the cal realist model is a mixture of the models corre- phenomenon. sponding to the vertices. Such a mixture is a hidden Note that even if the experiment is repeated a variables model, the hidden variable being the par- large number of times, it can never prove that prob- ticular random vertex chosen by the mixing distri- abilities like Pr A B are exactly equal to 1. It bution in a specific instance. { ≤ } STATISTICS, CAUSALITY AND BELL’S THEOREM 13

From quantum mechanics, after we have fixed a combined with an example of a no-signalling model joint p-partite quantum state, and sets of q r-valued which violates Tsirelson’s inequality). measurements per party, we will be able to write Let us investigate the local-realist models in more down probability tables p(a, b, . . . x,y,...) where the detail. A special class of local-realist models are variables x, y, etc. take values in| 1,...,q, and label the local-deterministic models. A local-deterministic the measurement used by the first, second, . . . party. model is a model in which all of the probabilities The variables a, b, etc., take values in 1, . . . , r and la- p(a, b, . . . x,y,...) equal 0 or 1 and the no-signalling | bel the possible outcomes of the measurements. Al- constraints are all satisfied. This implies that for together, there are qprp “elementary probabilities” each possible measurement by each party, the out- in this list of tables. More generally, any specific come is prescribed, independently of what measure- instance of a theory, whether local-realist, quantum ments are made by the other parties. Now, it is easy mechanical, or beyond, generates such a list of prob- to see that any local-realist model corresponds to ability tables, and defines thereby a point in qprp- a probability mixture of local-deterministic models. dimensional Euclidean space. After all, it “is” a joint probability distribution of si- We can therefore envisage the sets of all local- multaneous outcomes of each possible measurement realist models, all quantum models, and so on, as on each system, and thus it “is” a probability mix- subsets of qprp-dimensional Euclidean space. Now, ture of degenerate distributions: fix the random ele- whatever the theory, for any values of x, y, etc., ment ω, and each outcome of each possible measure- the sum of the probabilities p(a, b, . . . x,y,...) must ment of each party is fixed; we recover their joint equal 1. These are called normalisation| constraints. distribution by picking ω at random. Moreover, whatever the theory, all probabilities This makes the set of local-realist models a con- vex polytope: all mixtures of a finite set of extreme must be nonnegative: positivity constraints. Quan- points. Therefore, it can also be described as the in- tum mechanics is certainly local in the sense that tersection of a finite collection of half-spaces, each the marginal distribution of the outcome of any one half-space corresponding to a boundary hyperplane. of the measurements of any one of the parties does It can also be shown that the set of quantum mod- not depend on which measurements are performed els is closed and convex, but its boundary is very by the other parties. Since marginalization corre- difficult to describe. sponds again to summation of probabilities, these Let us think of these three models from “within” so-called no-signalling constraints are expressed by the affine sub-space of no-signalling and normali- linear equalities in the elements in the probability sation. Relative to this sub-space, the no-signalling tables corresponding to a specific model. Not sur- models form a full (nonempty interior) closed con- prisingly, local-realist models also satisfy the no- vex polytope. The quantum models form a strictly signalling constraints. smaller closed, convex, full set. The local-realist We will call a list of probability tables restricted models form a strictly smaller still, closed, convex, only by positivity, normalisation and no-signalling, full polytope. but otherwise completely arbitrary, a no-signalling Slowly, we have arrived at a rather simple picture, model. The positivity constraints are linear inequal- Figure 3. Imagine a square, with a circle inscribed in ities which place us in the positive orthant of Eu- it, and with another smaller square inscribed within clidean space. Normalisation and no-signalling are the circle. The outer square represents the bound- linear equalities which place us in a certain affine ary of the set of all no-signalling models. The circle sub-space of Euclidean space. Intersection of orthant is the boundary of the convex set of all quantum and affine sub-space creates a convex polytope: the models. The square inscribed within the circle is the set of all no-signalling models. We want to study the boundary of the set of all local-realist models. The sets of local-realist models, of quantum models, and picture is oversimplified. For instance, the vertices of of no-signalling models. We already know that local- the local-realist polytope are also extreme points of realist and quantum are contained in no-signalling. the quantum body and vertices of the no-signalling It turns out that these sets are successively larger, polytope. and strictly so: quantum includes all local-realist and A generalised Bell inequality is simply a bound- more (that’s Bell’s theorem); no-signalling includes ary hyperplane, or face, of the local-realist poly- all quantum and more (that is Tsirelson’s inequality tope, relative to the normalisation and no-signalling 14 R. D. GILL

Fig. 3. Left: a caricature of the 2 × 2 × 2 case. It actually lives in 8, not 2 dimensions. Right: caricature of the general case in which (bottom left) a further possibility is allowed: no purple between the green and grey. Artwork: Daniel Cavalcanti. affine sub-space, and excluding boundaries corre- Much of the material of this section is covered in sponding to the positivity constraints. I will call an excellent survey paper by Brunner et al. (2014), these interesting boundary hyperplanes “nontriv- from which I took, with the permission both of the ial”. In the 2 2 2 case, for which the affine authors and of the artist Daniel Cavalcanti, the two sub-space where× all× the action lies is 8-dimensional, illustrations in Figure 3: the first is a cartoon of the the local-realist polytope has exactly 8 nontrivial 2 2 2 case, the second of the general case. × × boundary hyperplanes. They correspond exactly to all possible CHSH inequalities (obtained by permut- 9. QUANTUM RANDI CHALLENGES ing outcomes, measurements and parties). Thus, in A second reason for the specific form of the proof the 2 2 2 case, the Bell-CHSH inequality is in- × × of Bell’s theorem which started this paper is that deed “all there is”. it lends itself well to design of computer challenges. When we increase p, q or r, new Bell inequali- Every year, new researchers publish, or try to pub- ties turn up, and moreover, keep turning up (“new” lish, papers in which they claim that Bell made means not obtainable from “old” by omitting parties some fundamental errors, and in which they put for- or measurements or grouping outcomes). It seems a ward a specific local realist model which allegedly hopeless (and probably pointless) exercise to try to reproduces the quantum correlations. The papers classify them. are long and complicated; the author finds it hard A natural question is whether every nontrivial to get the work published, and suspects a conspir- generalised Bell inequality can actually be violated acy by “The Establishment”. The claims regularly by quantum mechanics. I posed this as an open ques- succeed in attracting media attention, occasionally tion a long time ago, and for a long time the answer becoming head-line news in serious science journal- seemed probably be “yes”. However, a nice counter- ism; some papers are published, too, and not only example has recently been discovered; see Almeida in obscure journals. et al. (2010). Extraordinary claims require extraordinary evi- Quite a few generalised Bell inequalities have dence. I used to find it useful in debates with “Bell- turned out to be of particular interest, for instance, deniers” to challenge them to implement their local the work of Zohren and Gill concerned the 2 2 r realist model as computer programs for a network × × case and discussed a class of inequalities, one for of classical computers, connected so as to mimic the each r, whose asymptotic properties could be stud- time and space separations of the Bell-CHSH exper- ied as r increased to infinity. Further statistical con- iments. nections to missing data problems and optimal ex- The protocol of the challenge I issued in the past is perimental design, have been exploited by van Dam, the following. Bell-denier is to write computer pro- Gill and Gr¨unwald (2005) and Gill (2007). grams for three personal computers, which are to STATISTICS, CAUSALITY AND BELL’S THEOREM 15 play the roles of source , measurement station , Sascha Vongehr’s QRC completely cuts out any and measurement stationS . The following is toA be necessity for communication, protocol verification, repeated say 15,000 times.B First, sends messages adjudication. In fact, the Bell-denier no longer has to and . Next, connections betweenS , and to cooperate with myself or with any other member areA severed.B Next, from the outside worldA B so to of the establishment. They simply have to write a speak,S I deliver the results of two coin tosses (per- program which should perform a certain task. They formed by myself), separately of course, as input post their program on internet. If others find that it setting to and to . Heads or tails correspond to does indeed perform that task, the news will spread a request forA A or A′Bat , and for B or B′ at . The like wildfire. two measurement stationsA and now eachB output Vongehr prefers Bell’s original inequality, and I an outcome 1. Settings andA outcomesB are collected prefer CHSH, so I will here present an (unautho- for later data± analysis, Bell-denier’s computers are rised) “CHSH style” modification of his QRC. re-connected; next run. Suppose someone has invented a local hidden vari- Bell-denier’s computers can contain huge tables of ables theory. He can use it to simulate N =800 runs random numbers, shared between the three, and of of a CHSH experiment. Typically, he will simulate course they can use pseudo-random number gener- the source, the photons, the detectors, all in one ators of any kind. By sharing the pseudo-random program. Let us suppose that his computer code keys in advance, they have resources to any amount produces reproducible results, which means that the of shared randomness they like. code or the application is reasonably portable, and In Gill (2003), I showed how a martingale Hoeffd- will give identical output when run on another com- ing inequality gives an exponential bound like (3) puter with the same inputs. In particular, if it makes in the situation just described. This enabled me to use of a pseudo random number generator (RNG), choose N, and a criterion for win/lose (say, halfway it must have the usual “save” and “restore” facilities between 2 and 2√2), and a guarantee to Bell-denier for the seed of the RNG. Let us suppose that the pro- (at least so many runs with each combination of set- gram calls the RNG the same number of times for tings), such that I would happily bet 3000 Euros any each run, and that the program does not make use day that Bell-denier’s computers will fail the chal- in any way of memory of past measurement settings. lenge. The program must accept any legal stream of pairs The point (for me) was not to win money for my- of binary measurement settings of any length N. self, but to enable the Bell-denier who considers ac- In particular then, the program can be run with cepting the challenge (a personal challenge between N = 1 and all four possible pairs of measurement the two of us, with adjudicators to enforce the pro- settings, and the same initial random seed, and it tocol) to discover for him or herself that “it can- will thereby generate successively four pairs (A, B), not be done”. It is important that the adjudicators (A′,B), (A, B′), (A′,B′). If the programmer neither do not need to look inside the programs written by cheated nor made any errors, in other words, if the the Bell-denier, and preferably do not even need to program is a correct implementation of a genuine look inside his computers. They are black boxes. The LHV model, then both values of A are the same, only thing that has to be enforced are the commu- and so are both values of A′, both values of B, and nication rules. However, there are difficulties here. both values of B′. We now have the first row of the What if Bell-denier’s computers are using a wireless N 4 spreadsheet of Section 2 of this paper. network which the adjudicators cannot detect? The× random seed at the end of the previous phase A new kind of computer challenge, called the is now used as the initial seed for another phase, the “quantum Randi challenge”, was proposed in 2011 second run, generating a second row of the spread- by Sascha Vongehr (Science2.0: QRC). It is inspired sheet. This is where the prohibition of exploiting by the well known challenge to “paranormal phe- memory comes into force. The second row of coun- nomena” by James Randi (scientific sceptic and terfactual outcomes has to be completed without fighter against pseudo-science, see Wikipedia: James knowing which particular setting pair Alice and Bob Randi). Vongehr’s challenge (see Vongehr, 2012, will actually pick for the first row. 2013) differs in a number of fundamental respects Notice that the LHV model is allowed to use time, from mine, which indeed was not a quantum Randi since the saved random seeds could also include the challenge in Vongehr’s sense. current run number and the initial random seed 16 R. D. GILL value, too: in other words, when doing the calcu- APPENDIX: PROOF OF THEOREM 1 lations for the nth run, the LHV model has access The proof of (3) will use the following two Hoeffd- to everything it did in the previous n 1 runs. My claim is that a correct implementation− of a ing inequalities: bona-fide LHV model which does not exploit the Fact 3 (Binomial). Suppose X Bin(n,p) and memory loophole can be used to fill in the N 4 t> 0. Then ∼ spreadsheet of Section 2. When we now generate× Pr(X/n p + t) exp( 2nt2). random settings and calculate the correlations, we ≥ ≤ − get the same results as if they had been submitted Fact 4 (Hypergeometric). Suppose X is the in a single stream to the same program, run once number of red balls found in a sample without re- with the same initial seed. placement of size n from a vase containing pM red My new CHSH-style QRC to any local realist balls and (1 p)M blue balls and t> 0. Then out there who is interested, is that they program − their LHV model, modified so that it simply ac- Pr(X/n p + t) exp( 2nt2). cepts a random seed and value of N, and outputs an ≥ ≤ − N 4 spreadsheet. They should post it on internet Proof of Theorem 1. In each row of our N × 4 table of numbers 1, the product AB equals 1.× and draw attention to it on any of the many in- ± ± ternet fora devoted to discussions of quantum foun- For each row, with probability 1/4, the product is obs dations. Anyone interested runs the program, gener- either observed or not observed. Let NAB denote the ates N 2 settings, and calculates CHSH. If the pro- number of rows in which both A and B are observed. gram reproducibly,× repeatedly (significantly more Then N obs Bin(N, 1/4), and hence by Fact 3, for AB ∼ than half the time, cf. Conjecture 1 of Section 2), any δ> 0, violates CHSH, then the creator has created a classi- N obs 1 cal physical system which systematically violates the Pr AB δ exp( 2Nδ2). CHSH inequalities, thereby disproving Bell’s theo-  N ≤ 4 −  ≤ − rem. No establishment conspiracy can stop this news Let N + denote the total number of rows (i.e., AB − from spreading round the world, everyone can repli- out of N) for which AB = +1, define N simi- cate the experiment. The creator will get the No- AB larly. Let N obs,+ denote the number of rows such bel prize and there will be incredible repercussions AB that AB = +1 among those selected for observation throughout physics. obs obs,+ Some local realists will however insist on using of A and B. Conditional on NAB = n, NAB is dis- memory. They cannot rewrite their programs to tributed as the number of red balls in a sample with- create one N 4 spreadsheet. Instead, N rounds out replacement of size n from a vase containing × + − of communication are needed between themselves N balls of which NAB are red and NAB are blue. obs and some trusted neutral vetting agency. To bor- Therefore by Fact 4, conditional on NAB = n, for row an idea I learnt from Han Geurdes, we should any ε> 0, think of some kind of rating agency such as those obs,+ N N + for banks, an independent agency which carries out AB AB 2 Pr obs + ε exp( 2nε ). “stress tests”, on demand, but at a reasonable price,  NAB ≥ N  ≤ − to anyone who is interested and will pay. The pro- Recall that AB stands for the average of the cedure is almost as before: it ensures yet again that product AB overh thei whole table; this can be rewrit- the LHV model is legitimate, or more precisely, is ten as legitimate in its implemented form. The agency gen- − N + N N + erates a first run of settings (i.e., one setting pair), AB = AB − AB = 2 AB 1. but keeps it secret for the moment. The LHV theo- h i N N − rist supplies a first run-set of values of (A, A′,B,B′). Similarly, AB obs denotes the average of the prod- The agency reveals the first setting pair, the LHV h i ′ ′ uct AB just over the rows of the table for which theorist generates a second run set (A, A ,B,B ). both A and B are observed; this can be rewritten This is repeated N = 800 times. The whole proce- as dure can be repeated any number of times, the re- obs,+ obs,− obs,+ sults are published on internet, everyone can judge N N N AB = AB − AB = 2 AB 1. for themselves. obs obs obs h i NAB NAB − STATISTICS, CAUSALITY AND BELL’S THEOREM 17

obs obs obs For given δ> 0 and ε> 0, all of NAB , NAB′ , NA′B Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V. obs 1 and Wehner, S. (2014). Bell nonlocality. Rev. Modern and NA′B′ are at least ( 4 δ)N with probability at least 1 4 exp( 2Nδ2). On− the event where this Phys. 86 419; Erratum Rev. Modern Phys. 86 839. happens,− the conditional− probability that AB Christensen, B. G., McCusker, K. T., Altepeter, J., obs Calkins, B., Gerrits, T., Lita, A., Miller, A., exceeds AB + 2ε is bounded by h i h i Shalm, L. K., Zhang, Y., Nam, S. W., Brunner, N., exp( 2N obsε2) exp( 2( 1 δ)Nε2). Lim, C. C. W., Gisin, N. and Kwiat, P. G. (2013). − AB ≤ − 4 − Detection-loophole-free test of quantum nonlocality, and The same is true for the other three averages (for the applications. Phys. Rev. Lett. 111 130406. last one we first exchange the roles of + and to Cirel’son, B. S. (1980). Quantum generalizations of Bell’s 4 get a bound on A′B′ ). Combining everything,− inequality. Lett. Math. Phys. 93–100. MR0577178 h− iobs Clauser, J. F., Horne, M. A., Shimony, A. and we get that Holt, R. A. (1969). Proposed experiment to test local ′ ′ ′ ′ 49 AB + AB + A B A B 2 + 8ε, hidden-variable theories. Phys. Rev. Lett. 1804–1806. h iobs h iobs h iobs − h iobs ≤ Einstein, A., Podolsky, B. and Rosen, N. (1935). Can except possibly on an event of probability at most quantum-mechanical description of reality be considered complete? Phys. Rev. 47 777–780. 2 1 2 p = 4 exp( 2Nδ )+4exp( 2( δ)Nε ). Gallicchio, J., Friedman, A. S. and Kaiser, D. I. (2014). − − 4 − 2 Testing Bell’s inequality with cosmic photons: Closing We want to bound p by 8 exp( N(η/16) ) where the setting-independence loophole. Phys. Rev. Lett. 112 2 2 − 2 2 η = 8ε, making (η/16) = (ε/2) . Choosing 8δ = ε , 110405. 2 2 2 1 Gill, R. D. we find 2δ = (ε/2) = (η/16) . If 8( 4 δ) 1, then (2009). Schr¨odinger’s cat meets Occam’s razor. p 8 exp( 2Nδ2) and we are home. 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